By Claim REF  the density
of \(\mathcal {S}_k\)  is slightly below \(3/2\) , and hence, for \(p\ge n^{-2/3}\)  we know that with high probability a random graph
\(G\sim G_{n,p}\)  contains copies of \(\mathcal {S}_k\) . (See e.g. [1]} for subgraph containment in \(G_{n,p}\) .) Moreover, increasing slightly to \(p \ge n^{-2/3+\varepsilon }\)  and having that \(k=k(\varepsilon )\)  is a sufficiently large constant, it can be verified that in expectation
for any two vertices \(v,x\in V(G)\)  there is a copy of \(\mathcal {S}_k\)  such that \(v\)  is the copy of \(s_{k,1}\)  and \(x\)  is the copy of \(s_0\) . Now, throughout the game, Walker aims to find such copies which additionally are free of Breaker's edges and which can be reached from her current position \(a\)  within one round.
In order to guarantee that Walker can indeed find such copies, we want to make sure that there exists a collection of many copies of \(\mathcal {S}_k\)  with the additional property that any Breaker edge can only belong to a comparatively small number of these copies. This property may help later to ensure that Walker can play without facing a situation in which she wants to add a new vertex \(x\)  to her graph but cannot find a good structure with which she can still reach \(x\) .
